goodyear2013 wrote:
Merry and Michelle play a card game. In the beginning of the game they have an equal number of cards. Each player, at her turn, gives the other a third of her cards. Michelle plays first, giving Merry a third of her cards. Merry plays next, and Michelle follows. Then the game ends. Merry ended up with 14 more cards than Michelle. How many cards did each player have originally?
A) 18
B) 27
C) 36
D) 45
E) 54
Let x = the number of cards they each have at the beginning of the game.
After the first turn, i.e., after Michelle gives one-third of her cards (i.e., (1/3)x cards) to Merry, Michelle has x - (1/3)x = (2/3)x cards and Merry has x + (1/3)x = (4/3)x cards.
After the second turn, i.e., after Merry gives one-third of her cards (i.e., (1/3)(4/3)x = (4/9)x cards) to Michelle, Merry has (4/3)x - (4/9)x = (8/9)x cards and Michelle has (2/3)x + (4/9)x = (10/9)x cards.
After the third (or last) turn, i.e., after Michelle gives one-third of her cards (i.e., (1/3)(10/9)x = (10/27)x cards) to Merry, Michelle has (10/9)x - (10/27)x = (20/27)x cards and Merry has (8/9)x + (10/27)x = (34/27)x cards.
Since Merry ended up with 14 more cards than Michelle, we subtract the number of Michelle’s cards from the number of Merry’s cards, obtaining:
(34/27)x - (10/27)x = 14
(14/27)x = 14
x = 14 * (27/14)
x = 27
Answer: B
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